The hypergeometric functions that correspond to Groups 1 and 2 have z as
variable. The hypergeometric functions that correspond to Groups 3 and 4 have
a nonlinear function of z as variable. The transformation formulas between
two hypergeometric functions in Group 2, or two hypergeometric functions in
Group 3, are the linear transformations (15.8.1).

In the equations that follow in this subsection all functions take their
principal values.

provided that z lies in the intersection of the open disks
|z-14±14⁢3⁢i|<12⁢3,
or equivalently, |ph⁡((1-z)/(1+2⁢z))|<π/3. This is used in a
cubic analog of the arithmetic-geometric mean. See Borwein and Borwein (1991),
and also Berndt et al. (1995).

For further examples and higher-order transformations see
Goursat (1881), Watson (1910), and
Vidūnas (2005); see also
Erdélyi et al. (1953a, pp. 67 and 113–114).